10 INTEGRALS OF VECTOR-VALUED FUNCTIONS I57 compact (by proving that it is precompact) and coincides with the set of barycenters bM of positive measures with total mass 1 and support contained m H. (Remark that as ft runs through the set P <= M+(H) of positive measures with total mass 1, the mapping ft,h-»bM is continuous with respect to the vague topology on P and the topology induced on K by the product topology on RE/, and use Section 13.4, Problem 12.) 3. Let E be a separable real Frechet space and A a closed convex subset of E. Show that every convex function /(Section 13.9, Problem 14) which is lower semicontinuous on A is the upper envelope (12.7) of a family of functions which are the restrictions to A of continuous affine linear functions on E (i.e., functions of the form ZK-># + «(z), where u is a continuous linear form on E, and a e R). (Apply the Hahn-Banach theorem to the closed convex set of points (z, £) e E x R such that z e A and £ ^/(z).) 4. Let E be a separable real Frechet space, K a compact convex subset of E, /x a positive measure of total mass 1 on K, and bM the barycenter of ft (Problem 2). Show that if / is a convex real-valued function §; 0 which is lower semicontinuous on K, then J* fdp> (use Problem 3). 5. (a) Let E be a separable Frechet space, K a compact convex subset of E, and z a point of K. Show that every positive measure ft on K, with total mass 1, which has z as barycenter, lies in the vague closure of the set of positive measures of total mass 1 and finite support which have z as barycenter. (Let U be a neighborhood of ft with respect to the vague topology, consisting of measures v on K such that |/x(/i) — v(/f)| ^ 3 for some functions J\ e ^C(K). Show that there exists a finite number of points a, e K, and for each j a compact convex neighborhood \Y/ of a^ in K, such that the W/ cover K and |/,(y) —/t(aj)| g JS for each / and each y e XV,. Show (by using a continuous partition of unity) that we can write p in the form]T ocj ft;, where each ft/ is a positive measure of total mass 1 and support contained in Wj, and each a/ ^ 0 and £ a/ = 1. j If Zj is the barycenter of ft/, show that the barycenter of v =]T otj ez is z, and that v e U.) j (b) Let Kx be a compact subset of K, such that K is the closed convex hull of K/. For a point z e K to be an extremal point of K (Section 12.15, Problem 5) it is necessary and sufficient that e2 should be the only positive measure on K' with total mass 1 and z as barycenter. (Use (a) to prove that the condition is necessary.) 6. Let E be a separable real Frechet space and K a compact convex subset of E. Show that there exists a strictly convex continuous real-valued function on K. (The set J/ of re- strictions to K of continuous affine linear functions on E is a subspace of the separable metrizable space ^(K), hence contains a dense sequence (/*„). Show that, for suitable scalars an>0, the function ]T an hi satisfies the required conditions, by using the n Hahn-Banach theorem.) 7. Let E be a separable real Frechet space, K a compact convex subset of E; let u be a convex continuous real-valued function on K, let G ^ E x R be the graph of w, and let S be the closed convex hull of G in E x R. (a) Let v be the lower envelope of the continuous affine linear functions w on E such that w(z) g> w(z) for all z e K. Show that S is the set of points (z, £) e E x R such thateach n a measure v e K such that